Master Surface Area & Volume: Cones, Spheres & Hemispheres Guide

📐 Mastering 3D Shapes: Surface Area & Volume Guide

The Right Circular Cone, Surface Area of a Sphere, Volume Formulas, Worked Examples.

1. The Right Circular Cone 🍦

The Formulas

CSA (Curved Area)	πrl
TSA (Total Area)	πr(l + r)
Volume	(1/3)πr²h

⚠️ The Trap: If you need "l" but only have "h" and "r", use Pythagoras:
l² = r² + h²

📝 Example 1: Finding Curved Surface Area

Problem: A corn cob shaped like a cone has a radius (r) of 2.1 cm and a height (h) of 20 cm. Find the Curved Surface Area.

Step 1: Find the missing Slant Height (l).
We have h and r, but the formula (πrl) needs l.
l = √(2.1² + 20²) = √(4.41 + 400) = √404.41
l ≈ 20.11 cm

Step 2: Calculate Area.
CSA = (22/7) × 2.1 × 20.11
Answer: 132.73 cm²

📝 Example 2: Finding Volume

Problem: Calculate the volume of a cone with a base radius of 7 cm and a height of 10 cm.

Step 1: Select Formula.
Volume = (1/3)πr²h

Step 2: Calculate.
V = (1/3) × (22/7) × (7)² × 10
V = (1/3) × 22 × 7 × 10
V = 1540 / 3
Answer: 513.33 cm³

2. The Sphere ⚽

The Formulas

Surface Area	4πr²
Volume	(4/3)πr³

📝 Example 3: Surface Area of a Sphere

Problem: Find the surface area of a ball with a radius (r) of 7 cm.

Step 1: Apply Formula (4πr²).
Area = 4 × (22/7) × 7 × 7

Step 2: Solve.
The 7 in the denominator cancels out one 7.
Area = 4 × 22 × 7 = 88 × 7
Answer: 616 cm²

3. The Hemisphere 🥣

Half a sphere. Remember, it has a curved bottom and a flat top!

The Formulas

Curved SA	2πr²
Total SA	3πr²
Volume	(2/3)πr³

📝 Example 4: Total Surface Area

Problem: Find the total surface area of a solid hemisphere with a radius of 21 cm.

Step 1: Choose Formula.
Since it is "solid," we need the curved part AND the flat top.
Formula = 3πr²

Step 2: Calculate.
TSA = 3 × (22/7) × 21 × 21
TSA = 3 × 22 × 3 × 21
Answer: 4158 cm²

📝 Example 5: Volume of a Bowl

Problem: How much water can a hemispherical bowl hold if its radius is 3.5 cm?

Step 1: Choose Formula.
Volume = (2/3)πr³

Step 2: Calculate.
V = (2/3) × (22/7) × 3.5 × 3.5 × 3.5
V ≈ 0.66 × 3.14 × 42.875
Answer: 89.8 cm³

🎓 Summary Checklist

• Identify the Shape: Is it a Cone, Sphere, or Hemisphere?
• Surface Area vs. Volume: Are you wrapping it (Area) or filling it (Volume)?
• Check Variables: Do you need Slant Height ($l$)? Remember Pythagoras!
• Don't forget the Lid: For Hemispheres, is it an open bowl ($2\pi r^2$) or a solid object ($3\pi r^2$)?

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📚 Master Geometry • Examples based on Course PDF